We prove that D0, the set of all parameters λ ∈ C \ {0} for which the cubic polynomial fλ is parabolic and has no other parabolic or finite attracting periodic cycles, contains a deleted neighborhood of the origin 0. Our main result is that the function D0 3 λ 7→ HD(J(fλ)) ∈ R is real-analytic. This function ascribes to the polynomial fλ the Hausdorff dimension of its Julia set J(fλ). The theor...